Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, solve each system by Gaussian elimination. $$ \begin{array}{c} x+y+z=3 \\ \frac{x-1}{2}+\frac{y-3}{2}+\frac{z+1}{2}=0 \\ \frac{x-2}{3}+\frac{y+4}{3}+\frac{z-3}{3}=\frac{2}{3} \end{array} $$

Step-by-Step Solution

Answer

Infinite solutions: \( x+y+z=3 \), \( (3-y-t, y, t) \).

Step 1: Simplify Second Equation

Start by eliminating the fractions in the second equation: \[ \frac{x-1}{2} + \frac{y-3}{2} + \frac{z+1}{2} = 0 \]. Multiply the entire equation by 2 to get: \( x - 1 + y - 3 + z + 1 = 0 \), which simplifies to \( x + y + z = 3 \).

Step 2: Simplify Third Equation

Similarly, simplify the third equation by removing the fractions: \[ \frac{x-2}{3} + \frac{y+4}{3} + \frac{z-3}{3} = \frac{2}{3} \]. Multiply the equation by 3: \( x - 2 + y + 4 + z - 3 = 2 \), which simplifies to \( x + y + z = 3 \).

Step 3: Recognize the System Structure

Now we have the following system: \[ \begin{array}{c} x + y + z = 3 \ x + y + z = 3 \ x + y + z = 3 \end{array} \]. This shows that all equations are identical, providing infinite solutions along the line defined by \( x + y + z = 3 \).

Step 4: Express General Solution

To express the solution in parametric form, set one variable, for example, \( z = t \), where \( t \) is any real number. Then the solution becomes: \( x = 3 - y - t \) and \( y = y \). So, a general solution is \( (3 - y - t, y, t) \) where \( y \) and \( t \) are parameters.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple equations that share variables. In the example, the system is composed of three equations:

• Equation 1: \( x + y + z = 3 \)
• Equation 2: Simplified to \( x + y + z = 3 \)
• Equation 3: Simplified to \( x + y + z = 3 \)

Each equation represents a plane in 3-dimensional space. The solution to the system, when it exists, is the set of all points where these planes intersect.
In our case, all the equations describe the same plane. Therefore, the solution is where this plane intersects with any given line expressed as \( x + y + z = 3 \). This is what we refer to when we say the system has infinite solutions. Understanding systems of linear equations helps unravel complex problems in algebra by transferring real-world situations into mathematical terms.

Parametric Solutions

When a system of equations does not have a unique solution, parametric solutions become instrumental. Instead of presenting a definitive value for each variable, we express one or more variables in terms of parameters. In the exercise given:

• Set \( z = t \), where \( t \) is an unrestricted real number.
• The equation \( x + y + z = 3 \) then allows us to solve for \( x \) and \( y \) in terms of \( t \).

We derive:

• \( x = 3 - y - t \)
• \( y \) remains as itself since it can adjust based on \( t \).

This approach effectively demonstrates how such systems can be generalized to express a line, a plane, or a higher-dimensional analogue, all depending on available parameters. This can offer insight into how linear systems can extend into infinite sets of solutions.

Infinite Solutions

Infinite solutions occur when there is not just one unique solution to a system, but an entire set of values that will satisfy all the equations simultaneously. This is often visualized as overlapping lines or planes.

• For the system: \( x + y + z = 3 \), all three equations represent the same plane.
• Hazarding different values for one or two variables while still satisfying the equation gives multiple solution combinations.

Thus, the system is consistent but dependent, meaning it is impossible to pin down exact values of \( x, y, \) and \( z \) without further constraints.
In practical terms, these solutions imply flexibility and options, akin to having a solution path that can be customized depending on additional conditions or parameters such as \( z = t \). This provides a broader understanding of flexibility in mathematical modeling, which is particularly relevant in fields requiring adaptability.